Teaching Geometry
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This presentation compares the development of procedural fluency and conceptual understanding and argues for a systematic approach of teaching one before the other.
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Teaching Geometry
- 1. Teaching Geometry Procedural Fluency vs. Conceptual Understanding By John J. Gaines
- 2. Workshop Goals • Understand some of the issues of teaching Geometry • Teaching Procedural Fluency vs. Teaching Conceptual Understanding • Observe a demonstration of a strategy • Explore examples of teaching Geometry with the CCSSM • Participate in a hands-on activity
- 3. Issues of Teaching Geometry • Procedural Fluency and Conceptual Understanding • Both are important for developing mathematically proficient students • Which one should be taught first? • Should they be taught simultaneously? Procedural Fluency Conceptual Understanding
- 4. Issues of Teaching Geometry Who remembers this formula? 𝑑 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2 What concepts are addressed in this formula? • Subtraction • Addition • Exponent • Square Root • Coordinate Points • Distance
- 5. Issues of Teaching Geometry The Distance Formula 𝑑 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2
- 6. Issues of Teaching Geometry 𝐴 = 𝑙 ∙ 𝑤
- 7. Issues of Teaching Geometry
- 8. Issues of Teaching Geometry 𝑨 = 𝒍 ∙ 𝒘 𝑨 𝟏 = 𝟏𝟎 𝒄𝒎 ∙ 𝟗𝒄𝒎 = 𝟗𝟎 𝒄𝒎 𝟐 𝑨 𝟐 = 𝟔 𝒄𝒎 ∙ 𝟐 𝒄𝒎 = 𝟏𝟐 𝒄𝒎 𝟐 𝑨 = 𝑨 𝟏 + 𝑨 𝟐 𝑨 = 𝟗𝟎 𝒄𝒎 𝟐 + 𝟏𝟐 𝒄𝒎 𝟐 𝑨 = 𝟏𝟎𝟐𝒄𝒎 𝟐
- 9. Issues of Teaching Geometry
- 10. Hands-On Activity For each of the shapes provided, respond to the following prompts: 1. Sketch the geometric shape and label its measurements. 2. What other shapes seem to make up this geometric shape? 3. How do you think you would find the area of this shape? 4. What formulas seem like they would be important for finding the area of this shape? 5. What is the area of this shape? Show how you found the area. 6. Where would you see this shape in the real world? 7. When could you be asked to find the area of this shape in the real world?
- 17. Implementation of CCSSM Grade 7 – Geometry • CSS.Math.Content.7.G.B.6 - Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Standards for Mathematical Practice • MP1 - Make sense of problems and persevere in solving them. • MP7 - Look for and make use of structure
- 18. Conclusion National Council of Teachers of Mathematics (NCTM) • “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems” (2014, p. 42). van Hiele Model of Geometric Thought • Level 0 – Visualization “The products of thought at level 0 are classes or groupings of shapes that seem to be ‘alike.” (Van de Walle, Karp, Bay-Williams, 2013, p. 403). • Level 1 – Analysis “The products of thought at level 1 are the properties of shapes” (Van de Walle et al., 2013, p. 405).
- 19. Conclusion • Procedures should be learned over time on a foundation of conceptual understanding. • In other words, developing conceptual understanding should precede developing procedural fluency. Conceptual Understanding Procedural Fluency
- 20. Resources Mathematics Assessment Project http://map.mathshell.org/ NCTM Illuminations https://illuminations.nctm.org/ National Library of Virtual Manipulatives http://nlvm.usu.edu/ Geometer’s Sketchpad http://www.dynamicgeometry.com/ Lesson Plans and Activities that focus on developing Conceptual Understanding Virtual manipulatives and online geometric design platforms used to facilitate conceptual understanding.
- 21. References California Department of Education. (2014). California Common Core State Standards for Mathematics. Retrieved September 26, 2015, from http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug201 3.pdf National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics. Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally. (8th ed.). Boston, MA: Pearson Education.
Editor's Notes
- Welcome, I am John J. Gaines, and I am here to talk about Teaching Geometry, with an emphasis on procedural fluency and conceptual understanding.
- Let’s look at some of the goals for this workshop. <click> First, we will look at some of the issues of teach Geometry, particularly focusing on when to teach procedural fluency and when to teach conceptual understanding. <click> Second, you will have the opportunity to observe one of the strategies discussed in this workshop. <click> Third, we will explore several examples of teaching Geometry with regards to the Common Core State Standards for Mathematical Content and for Mathematical Practice. <click> Lastly, we will finish with an activity for all of you to participate in, followed by a quick reflection.
- <click> One of the major issues in teaching Geometry is the notion of teaching to develop procedural fluency and conceptual understanding. <click> Both of these are important for developing mathematically proficient students, but how should they be taught? <click x 2> For years, instruction has been algorithmic, with the intention of developing conceptual understanding inherently through the development of procedural fluency. Over the years, however, this method of instruction has been questioned. Do students really develop conceptual understanding through developing procedural fluency or could they learn Geometry a different way? Procedural fluency is more than being able to carry out a certain mathematical procedure. Students must know what is the appropriate procedure to use, when the procedure should be used, and what results they should expect from using it. These go beyond just being able to carry out the procedure and require a conceptual understanding on the procedural in context.
- <click> Who remembers this formula? Some of you may remember this from school. You’re usually shown a line segment on an XY coordinate plane and asked to find the length of that line segment. Sometimes it’s connected to a right triangle as the hypotenuse. Still, when you see this formula, there is a lot of information to digest. This is especially true for our students. Before going on to the next slide, I would like you to take a moment to think about all of the concepts that addressed in the formula. <click> Then, I will have you turn to your neighbor and share what you came up with. [Hold a discussion for participants to share their thoughts.] Now consider this, the moment we write this formula on the board and ask our students to use it, we are asking them to understand all of the concepts that we just shared. <click x 6>
- When are we usually asked to use the distance formula? <click> It is usually when were shown a line segment, with some slope, drawn on the XY coordinate plane, like the one shown in this slide. Now, when you see the coordinate points of both endpoints of the line segment provided, knowing what you know, what are you inclined to do? (Possible answer: Plug the values in the distance formula and find the length of the line segment.) [Call on volunteers to answer the question.] Before we start plugging in values, let’s look at the picture a moment. Other than a random line segment, what do you see? I want you to think about this for a minute and share what you see in this picture with your neighbor. [Circulate as the participants share with their neighbors. Stop the conversations after a few minutes and call on volunteers to share with the rest of the group what they saw.] Looking at the picture here, did any of you see this? <click> There’s a right triangle here, with a base of 4 and a height of 3. Knowing this is a right triangle, we could probably use the Pythagorean Theorem to determine the length of the line segment (or the hypotenuse).
- How many of you have been shown this figure and asked to find its area? I am sure that many of you have. In fact, most of you are probably thinking about a formula that you could use to find the area. <click> The formula for the area of a rectangle is "A=l∙w” or the length <click> multiplied by the width <click>. I am sure that we have all been taught this, but what does it mean. If the length of a rectangle is 5 cm and the width is 3 cm, what does it mean to say that the area is 15 cm?
- If we look at a rectangle that is 5 𝑐𝑚 long by 3 𝑐𝑚 wide and divide it into 1𝑐𝑚×1𝑐𝑚 squares, we could count that there are 15 – 1𝑐𝑚×1𝑐𝑚 squares. Is this what area means? What if we looked at area in a different way and divided the 1𝑐𝑚×1𝑐𝑚 squares into groups. Instead of saying that the area of the rectangle is 15 𝑐 𝑚 2 , we could say that the area is equal to 5 groups of 3 squares each. <click x 5> We could also say that the area is equal to 3 groups of 5 squares. <click x 4>
- By only teaching students procedural fluency, we develop a culture of “plugging in” values without any consideration for using particular formulas or plugging in values for their (supposed) symbolic referents. For example, when presented with an irregular geometric shape that requires a more complicated approach to determining its area, some students will resort to the algorithm of multiplying length and width values that are adjacent to each other. <click x 6> This is actually a strategy that several of my students used when I first introduced irregular area to them. What do you notice about their solution? I want you to think about this for a minute and share what you noticed about the student’s solution with your neighbor. [Circulate as the participants share with their neighbors. Stop the conversations after a few minutes and call on volunteers to share with the rest of the group what they noticed.]
- We have just seen where teaching procedural fluency without an emphasis or a foundation in conceptual understanding could lead a student to incorrectly solving a problem involving an L-shaped quadrilateral. Now, what would happen if the same student was presented with these six geometric shapes? <click x 6> How would procedural fluency hold up in terms of these shapes? I want you to think about this for a minute and share what you think with your neighbor. [Circulate as the participants share with their neighbors. Stop the conversations after a few minutes and call on volunteers to share with the rest of the group what they think.] Now, that all of you have had some time to think about these shapes, you are going to participate in an activity involving these same shapes.
- In the packets that I have provided you, there are six different geometric shapes. I will divide you into six groups and have each group focus on a particular shape. [Divide the participants into six equal groups and assign them each a geometric shape.] For your assigned geometric shape, I want you and your group to respond to the following seven prompts: Sketch the geometric shape and label its measurements. What other shapes seem to make up this geometric shape? How do you think you would find the area of this shape? What formulas seem like they would be important for finding the area of this shape? What is the area of this shape? Show how you found the area. Where would you see this shape in the real world? When could you be asked to find the area of this shape in the real world? As you work on this, I will be circulating to observe and assist you in any way. [Circulate around the groups, observing and engaging the groups. After 15 to 20 minutes, call the attention of all the groups, and have them present their responses according to the geometric shapes that show up on the next six slides.]
- [Have the group of participants responsible for this geometric shape share their responses with the whole group. Click through the animations to either support or offer an alternative way to look at the shapes that make up this geometric shape.]
- [Have the group of participants responsible for this geometric shape share their responses with the whole group. Click through the animations to either support or offer an alternative way to look at the shapes that make up this geometric shape.]
- [Have the group of participants responsible for this geometric shape share their responses with the whole group. Click through the animations to either support or offer an alternative way to look at the shapes that make up this geometric shape.]
- [Have the group of participants responsible for this geometric shape share their responses with the whole group. Click through the animations to either support or offer an alternative way to look at the shapes that make up this geometric shape.]
- [Have the group of participants responsible for this geometric shape share their responses with the whole group. Click through the animations to either support or offer an alternative way to look at the shapes that make up this geometric shape.]
- [Have the group of participants responsible for this geometric shape share their responses with the whole group. Click through the animations to either support or offer an alternative way to look at the shapes that make up this geometric shape.]
- This activity that we just completed addresses both the Common Core State Standards for Mathematical Content and for Mathematical Practice. For example, this activity could have been part of a larger lesson on solving real-world and mathematical problems involving area of two-dimensional objects, relative to the 7th grade Mathematics standard in Geometry listed here. <click> [Read description of CSS.Math.Content.7.G.B.6.] This activity also addresses several Standards for Mathematical Practice, namely MP1 and MP7. <click> MP1 focuses on having the students make sense of the problem, while MP7 focuses on having the students use structure to make meaning of a problem. [Read description of both Standards for Mathematical Practice.]
- According to the National Council of Teachers of Mathematics <click x 2>, “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding” (2014, p. 42). In other words, it is their argument that students should conceptually understand mathematical problems before approaching their procedurally. In fact, the van Hiele model was developed on this notion as the initial levels of developing geometric reasoning and spatial reasoning focus on visualization and developing conceptual understanding.
- We have looked at different examples and have explored the consequences of teaching procedural fluency versus conceptual understanding. Undoubtedly both are crucial for developing mathematical proficient students. Teaching procedural fluency with the hope of students inherently developing conceptual understanding has created a disconnect between the students and the concepts of study. Consequently, they haphazardly use a procedure without understanding why they are using that procedure and plug in values that fit the procedure. Instead, we need to educate our students and develop their conceptual understanding first. By doing this, we will enable to perceive the different procedures as different entry points for approaching a problem, gaining a higher level of awareness as they work through a problem.
- I have divided the list of resources into two sections: (1) Lesson plans and activities and <click> (2) virtual manipulatives and online geometric design platforms. <click> Under lesson plans and activities, <click> I have listed the URLs for the Mathematics Assessment Project <click> and NCTM Illuminations. <click> Both of these websites offer an extensive amount of lesson plans and activity ideas for developing conceptual understanding. Under virtual manipulatives and online geometric design platforms, <click> I have listed the URLs for the National Library of Virtual Manipulatives <click> and the Geometer’s Sketchpad. <click> The National Library of Virtual Manipulatives offers a variety of virtual manipulatives to help develop conceptual understanding, including different styled geoboards. Geometer’s Sketchpad offers a way to represent two- and three-dimensional geometrical objects visually, helping students visualize the different objects.